A Note on the M/G/1 Queue with Server Vacations

1. A Note on the M/G/1 Queue with Server Vacations S.W. Fuhrmann Operations Research, Vol. 32, No. 6 (Nov. – Dec., 1984), 1368-1373 Chung, Chia-Fang

2. Outline • Problem Definition • Notation • The M/G/1 Queue with Server Vacations • Extensions Chung, Chia-Fang

3. Outline • Problem Definition • Notation • The M/G/1 Queue with Server Vacations • Extensions Chung, Chia-Fang

4. Problem Definition • Server begins a “vacation” of random length each time the system becomes empty. • If he returns to find no customers waiting, begins another vacation immediately. • If he returns to find customers waiting, works until the system empties and then begins another vacation. Chung, Chia-Fang

5. Problem Definition (cont’d) • The number of customers present in the system at a random point in time in equilibrium • The number of Poisson arrivals during a time interval that is distributed as the equilibrium forward recurrence time (residual life) of a vacation. • The number of customers present at a random point in time in equilibrium in the corresponding standard M/G/1 queuing system. Chung, Chia-Fang

6. Outline • Problem Definition • Notation • The M/G/1 Queue with Server Vacations • Extensions Chung, Chia-Fang

7. Notation • λ= the arrival rate of customers to the system • V(.) = the distribution function of the length of a vacation • v = the mean value of V(.) • R(.) = the distribution function of the equilibrium forward recurrence timeof a vacation (Cox and Lewis,1968. The Statistical analysis of Series of Events. Section 4.3) Chung, Chia-Fang

8. Notation (cont’d) • πi(.)=the p.g.f for the number of customers that a random departing customer leaves behind in system i. • Wi(.)= the distribution function of the sojourn time that a random customer experiences in system i, under a FIFO queuing discipline. Chung, Chia-Fang

9. Outline • Problem Definition • Notation • The M/G/1 Queue with Server Vacations • Extensions Chung, Chia-Fang

10. The M/G/1 Queue with Server Vacations • Assume that the lengths of vacations are i.i.d. and are independent of the arrival process and of the service times of customers. Chung, Chia-Fang

11. The M/G/1 Queue with Server Vacations (cont’d) • Remark 1 • πi(.)=Pi(.) • Remark 2 • Under a FIFO queuing discipline, the customers that a departing customer leaves behind are precisely those customers who arrived during the parting customer’s sojourn time. Chung, Chia-Fang

12. The M/G/1 Queue with Server Vacations (cont’d) Chung, Chia-Fang

13. Vacation Service Vacation The M/G/1 Queue with Server Vacations (cont’d) • Primary customers • Customers who arrive while the server is on vacation • Secondary customers • Customers who arrive while the server is busy • Virtual 1-busy period • The server begins service to a primary customer until the next time when that primary customer has departed and there are no secondary customers present in the system Chung, Chia-Fang

14. Vacation Busy Period Vacation Primary Customer No. of primary customers the tagged customer leaves behind The M/G/1 Queue with Server Vacations (cont’d) • Consider a random “tagged” customers as he departs from the system • Primary customers Chung, Chia-Fang

15. The M/G/1 Queue with Server Vacations (cont’d) • Secondary Customers • Arrives since the current virtual 1-busy period begins • Such virtual 1-busy period follows the same stochastic laws as in the system 2 • The number of customers the tagged customer leaves behind = π2(.) Chung, Chia-Fang

16. The M/G/1 Queue with Server Vacations (cont’d) • Total no. of customers the tagged customers left behind =primary customers + secondary customers independent Chung, Chia-Fang

17. Outline • Problem Definition • Notation • The M/G/1 Queue with Server Vacations • Extensions Chung, Chia-Fang

18. Extensions • To relax the assumptions for the following examples, • Models where each vacation ends precisely when a fixed number of primary customers are waiting • Cyclic queuing models in which the lengths of successive vacations are positively correlated. Chung, Chia-Fang

19. Notations • Xn= no. of primary customers present when the server returns from his nth vacation • P(Xn=k) = P(X=k) for all n, k • α(.)=p.g.f of X Chung, Chia-Fang

20. Notations (cont’d) • Consider a random primary customer. • X*=total no. of (primary) customers that arrive during current vacation. • P(X*=n) = nP(X=n) / α’(1) • Y=total no. of (primary) customers that arrive during current vacation but after the random primary customer arrived • β(.)=p.g.f of Y • P(Y=k | X*=n) = 1/n Chung, Chia-Fang

21. Proof • To prove π1(z) = β(z) π2(z) • No. of customers = the sum of two independent random variables • Find β(z) Chung, Chia-Fang

22. Proof (cont’d) Chung, Chia-Fang

23. Conclusion • The number of customers present in the system at a random point in time in equilibrium,π1(z) • The number of Poisson arrivals during a time interval that is distributed as the equilibrium forward recurrence time (residual life) of a vacation, • The number of customers present at a random point in time in equilibrium in the corresponding standard M/G/1 queuing system, π2(z) Chung, Chia-Fang